A proportion calculator solves the equation a/b = c/d when any three of the four values are known. A proportion states that two fractions (or rates) are equal, and cross multiplication turns that statement into a simple equation you can solve — the workhorse method behind recipe scaling, unit conversions, map reading, and similar-triangle problems.
Enter three values, leave the unknown blank, and the calculator finds it, displays the completed proportion, and verifies the answer by reducing both sides to the same decimal. That verification step is exactly how you should check proportion answers on homework and exams.
Cross Multiplication, Step by Step
If two fractions are equal, their cross-products are equal:
a/b = c/d ⟺ a × d = b × c
This works because multiplying both sides of a/b = c/d by b × d cancels each denominator. To solve for any single unknown:
- Unknown a: a = (b × c) ÷ d
- Unknown b: b = (a × d) ÷ c
- Unknown c: c = (a × d) ÷ b
- Unknown d: d = (b × c) ÷ a
Example: solve 3/4 = 9/d. Cross-multiply: 3 × d = 4 × 9 = 36, so d = 36 ÷ 3 = 12. Check: 3 ÷ 4 = 0.75 and 9 ÷ 12 = 0.75. Equal decimals confirm the proportion is true.
Setting Up Proportions from Word Problems
Most proportion mistakes happen during setup, not arithmetic. Keep the same units in the same positions on both sides:
- Recipe: "3 cups of flour make 24 cookies; how much flour for 40 cookies?" Set up flour/cookies = flour/cookies: 3/24 = x/40.
- Speed: "A car covers 150 miles in 3 hours; how far in 5 hours?" Miles/hours on both sides: 150/3 = x/5.
- Map scale: "1 inch represents 50 miles; how many miles is 3.5 inches?" Inches/miles: 1/50 = 3.5/x.
The golden rule: if the numerator on the left is flour, the numerator on the right must also be flour. Mixing positions — putting cookies on top on one side and flour on top on the other — produces a wrong equation that still looks plausible.
Worked Example: Scaling a Recipe
A cookie recipe uses 3 cups of flour to make 24 cookies. How much flour do you need for 40 cookies?
Step 1: Set up the proportion with matching units in matching positions: 3/24 = x/40 (flour over cookies on both sides). Step 2: Cross-multiply: 3 × 40 = 24 × x, giving 120 = 24x. Step 3: Divide both sides by 24: x = 120 ÷ 24 = 5.
You need 5 cups of flour. Verify by reducing both sides: 3 ÷ 24 = 0.125 cups per cookie, and 5 ÷ 40 = 0.125 cups per cookie. The per-cookie rate is identical, so the proportion — and the answer — is correct.
Frequently Asked Questions
How do you solve a proportion using cross multiplication?
Multiply the numerator of each fraction by the denominator of the other and set the products equal: a/b = c/d becomes a × d = b × c. Then divide to isolate the unknown. For 3/4 = 9/d, compute 3d = 36, so d = 12.
What is a proportion in math?
A proportion is an equation stating that two ratios or fractions are equal, such as 3/4 = 9/12. Both sides reduce to the same value (0.75 here). Proportions model any situation where two quantities change at the same rate, like price per item or miles per hour.
How do I check whether a proportion is true?
Use either test: (1) cross-multiply and see if the products match — for 3/4 = 9/12, both 3 × 12 and 4 × 9 equal 36; or (2) divide each fraction and compare decimals — 0.75 on both sides. If either test passes, the proportion is true.
How do I set up a proportion from a word problem?
Identify the two quantities being compared and keep each in the same position on both sides. If 150 miles pairs with 3 hours, write miles over hours on both sides: 150/3 = x/5. Label the units while setting up — mismatched positions are the most common source of wrong answers.
What is the difference between a ratio and a proportion?
A ratio compares two quantities (3:4), while a proportion is an equation saying two ratios are equal (3/4 = 9/12). You solve a proportion; you simplify or scale a ratio. Every proportion contains two ratios, and cross multiplication only applies to proportions.